The Schneider problem for a micropolar fluid

Fluid Dynamics Research 38 (2006) 489–502

The Schneider problem for a micropolar fluid

Anuar Ishaka, Roslinda Nazara, Ioan Popb,∗aSchool of Mathematical Sciences, National University of Malaysia, 43600 UKM Bangi, Selangor, Malaysia

bFaculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania

Received 21 October 2005; received in revised form 7 January 2006; accepted 13 March 2006

Communicated by M. Oberlack

Abstract

The effect of buoyancy forces on fluid flow and heat transfer over a horizontal plate in a steady, laminar andincompressible micropolar fluid has been investigated. The wall temperature is assumed to be inversely proportionalto the square root of the distance from the leading edge. The set of similarity equations has been solved numericallyusing the Keller-box method, and the solution is given for some values of buoyancy parameter, material (micropolar)parameter and Prandtl number. It is found that dual solutions exist up to certain negative values of buoyancy parameter(decelerated flow) for all values of micropolar parameter and Prandtl number considered in this study. Beyond thesevalues, the solution does no longer exist. Moreover, it is found that there is no local heat transfer at the wall exceptin the singular point at the leading edge, although the wall temperature is different from the free stream temperature.© 2006 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved.

Keywords: Boundary layer; Dual solutions; Heat transfer; Horizontal plate; Micropolar fluid; Mixed convection

1. Introduction

The micropolar model of fluid flow has attracted considerable attention from researchers since theformulation of the model by Eringen (1966, 1972), in which the local effect resulting from microstructureand gyration motions of the fluid elements was taken into consideration. The essence of the micropolartheory lies in the extension of the constitutive equations for Newtonian flow so that more complex

∗ Corresponding author. Tel.: +40 264 594 315; fax: +40 264 591 906.E-mail address: [email protected] (I. Pop).

0169-5983/$30.00 © 2006 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved.doi:10.1016/j.fluiddyn.2006.03.004

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490 A. Ishak et al. / Fluid Dynamics Research 38 (2006) 489–502

fluids can be studied. Examples of industrially relevant flows that can be studied using the theory in-clude the flow of low concentration suspensions, liquid crystals, ferro liquids, blood, porous media,dirty oil, lubrication, turbulent shear flow, etc. In practice, the theory requires that one must solve anadditional transport equation representing the principle of conservation of local angular momentum,as well as the usual transport equations for the conservation of mass and momentum, and additionalconstitutive parameters are also introduced. Extensive reviews of the theory and applications can befound in the review articles by Ariman et al. (1973, 1974) and the recent books by Łukaszewicz (1999)and Eringen (2001).

The potential importance of micropolar boundary layer flow in industrial applications has motivateda number of previous studies, of which those by Ahmadi (1976), Rees and Bassom (1996), Hassanien(1996) and Kim (1999) are most directly relevant to the present case where the effect of buoyancyforces on the steady micropolar boundary layer flow over a horizontal flat plate, taking into account thehydrostatic pressure is studied. It is known that the problem in which a uniform stream flows of a viscousand incompressible fluid (Newtonian fluid) past a flat plate at a uniform temperature above that of theambient fluid does not admit a similarity solution, see Merkin and Ingham (1987). However, Schneider(1979) has shown that if the plate temperature is allowed to vary like x−1/2 (x is the non-dimensionaldistance from the leading edge) a similarity solution is possible. Thus, following the work by Schneider(1979) an exact similarity solution is given in this paper for the mixed convection boundary layer flowof a micropolar fluid over a horizontal flat plate when the wall temperature is proportional to x−1/2.We call it the Schneider problem for a micropolar fluid. This paper is, therefore, the generalization ofSchneider’s (1979) classical problem for a Newtonian fluid to the micropolar fluids. This solution can beinterpreted to describe the mixed convection flow above a plate that is strongly heated or cooled at theleading edge while otherwise isolated. The resulting equations are different from those of both Schneider(1979) and Hassanien (1996). The outstanding feature of Schneider’s problem is that it is the only caseof a self-similar mixed convection boundary layer flow problems on a horizontal plate which can bereduced from a sixth- to a fourth-order boundary value problem. Following Rees and Bassom (1996),Rees and Pop (1998), and Nazar et al. (2003), we consider a special form of the spin gradient viscosity. Itis also assumed that the microinertia density is variable. Detailed numerical results are presented, as wellas an asymptotic analysis for large values of the Prandtl number similar to that presented by Steinrück(2001) for a Newtonian fluid. Such an analysis is necessary because the numerical results indicate that thesolution is not unique and that dual solutions exist for values of buoyancy parameter slightly larger than thecritical value and that the boundary layer approximation might break down. More detailed investigationsof breakdown of the boundary-layer approximation for mixed convection above a horizontal plate hasbeen given by Schneider and Wasel (1985).

2. Basic equations

Consider the steady flow of a micropolar fluid past a horizontal flat plate aligned parallel to a uniformfree stream with velocity U∞ and uniform temperature T∞, as shown in Fig. 1. A Cartesian coordinatesystem x, y is used with the origin at the leading edge of the plate and x- and y-axes measured alongthe plate and normal to it, respectively. The plate is maintained at a certain temperature Tw(x) andthe temperature of the ambient fluid is T∞. Under the boundary layer and Boussinesq approximations,

A. Ishak et al. / Fluid Dynamics Research 38 (2006) 489–502 491

Fig. 1. Physical model and coordinate system.

the governing equations can be written as (see Gorla, 1995; Rees and Bassom, 1996)

�u

�x+ �v

�y= 0, (1)

u�u

�x+ v

�u

�y=(

� + �

�

)�2u

�y2 + �

�

�N

�y+ g�

�

�x

[∫ ∞

y

(T − T∞) dy

], (2)

�j

(u

�N

�x+ v

�N

�y

)= �

�y

(��N

�y

)− �

(2 N + �u

�y

), (3)

u�j

�x+ v

�j

�y= 0, (4)

u�T

�x+ v

�T

�y= �

�2T

�y2 , (5)

where u and v are the velocity components along the x- and y-axes, respectively, N is the component ofthe microrotation vector normal to the x–y plane, T is the fluid temperature, g is the magnitude of theacceleration due to gravity, � is the thermal diffusivity, � is the thermal expansion coefficient, � is thedensity, � is the absolute viscosity, � is the vortex viscosity, � is the spin-gradient viscosity and j is themicroinertia density. We assume that the boundary conditions of Eqs. (1)–(5) are

u = v = j = 0, T = T w(x), N = −n�u

�yat y = 0,

u = U∞, T = T∞, N = 0 as y → ∞, (6)

where n is a constant such that 0�n�1. It should be mentioned that the condition n = 0, called strongconcentration by Guram and Smith (1980), is a generalization of the no slip condition, and indicatesN =0 near the wall. It represents concentrated particle flows in which the microelements close to the wallsurface are unable to rotate (Jena and Mathur, 1981). The case n �= 0 means that in the neighborhood ofa rigid boundary, the effect of microstructure is negligible since the suspended particles cannot get closer


to the boundary than their radius. Hence in the neighborhood of the boundary, the only rotation is dueto fluid shear and therefore, the gyration vector must be equal to fluid vorticity (Bhargava et al., 2003).Further, n = 1

2 indicates the vanishing of anti-symmetrical part of the stress tensor and denotes weakconcentration (Ahmadi, 1976). We shall consider here only the value of n = 1

2 (weak concentration).We follow the work of many recent papers (e.g. Rees and Bassom, 1996; Rees and Pop, 1998; Nazar

et al., 2003) by assuming that � is given by

� = (� + �/2)j = �(1 + K/2)j , (7)

where K = �/� is the material (micropolar) parameter.We introduce now the non-dimensional variables defined as

x = x

L, y = √

Re

(y

L

), u = u

U∞, v = √

Re

(v

U∞

),

N =(

L

U∞

)(N√Re

), T = T − T∞

T ∗ , j =(

U∞�L

)j , (8)

where Re=U∞L/� is the Reynolds number, L is a characteristic length, T ∗ is a characteristic temperatureand � is the kinematic viscosity. Eqs. (1)–(5) then become

�u

�x+ �v

�y= 0, (9)

u�u

�x+ v

�u

�y= (1 + K)

�2u

�y2 + K�N

�y+ �

�

�x

[∫ ∞

y

T dy

], (10)

j

(u

�N

�x+ v

�N

�y

)= (1 + K/2)

�

�y

(j

�N

�y

)− K

(2N + �u

�y

), (11)

u�j

�x+ v

�j

�y= 0, (12)

u�T

�x+ v

�T

�y= 1

Pr

�2T

�y2 , (13)

and the boundary conditions (6) become

u = v = j = 0, T = Tw(x), N = −n�u

�yat y = 0,

u = 1, T = 0, N = 0 as y → ∞, (14)

where Pr =�/� is the Prandtl number and � is the buoyancy parameter which is related to the Archimedesnumber Ar = g�T ∗L/U2∞ and Reynolds number according to the following relation:

� = Ar√Re

. (15)

It should be mentioned that for � > 0(T ∗ > 0), the flow is assisting, while for � < 0 (T ∗ < 0), the flow isopposing. We also notice that for K =0 (Newtonian fluid), Eqs. (9), (10) and (13) reduce to those derivedby Schneider (1979).


Following Schneider (1979) and Magyari et al. (2002), we assume that

Tw(x) = x−1/2, (16)

and introduce the following similarity variables:

(x, y) = x1/2f (), T (x, y) = x−1/2�(),

N(x, y) = x−1/2h(), j (x, y) = xi(), = y

x1/2 , (17)

where is the stream function defined in the usual way as u= �/�y and v =−�/�x, which identicallysatisfy Eq. (9).

Substituting (17) into Eqs. (10)–(13), we get the following ordinary differential equations:

2(1 + K)f ′′′ + ff ′′ + 2Kh′ + �� = 0, (18)

2(1 + K/2)(ih′)′ + i(f h′ + f ′h) − 2K(2h + f ′′) = 0, (19)

f i′ − 2f ′i = 0, (20)

2

Pr�′′ + f �′ + f ′� = 0, (21)

subject to the boundary conditions

f (0) = f ′(0) = i(0) = 0, �(0) = 1, h(0) = −nf ′′(0),f ′(∞) = 1, h(∞) = 0, �(∞) = 0. (22)

We notice that Eq. (21) can be integrated at once so that we get

2

Pr�′ + f � = 0, (23)

if the boundary condition �(∞) = 0 is used. It is interesting to point out that Eq. (23) gives �′(0) = 0 forall values of K, n and �. This result indicates that there is no local heat transfer at the plate surface for allx > 0. This situation is resolved by recalling that the similarity solution (17) requires a singular behaviorof the wall temperature at x = 0. Thus, all the heat necessary to change the fluid temperature must betransferred in the singular point x = 0, which is the leading edge of the plate (see Schneider, 1979). ThisEq. (23) with the boundary condition �(0) = 1 can be integrated to give

�() = exp

(−Pr

2

∫

0f d

). (24)

However, using Eq. (23) instead of Eq. (24) is more convenient for the numerical solution.The solution of Eq. (20) satisfying the boundary condition i(0)=0 and the boundary condition f (0)=0

is given by

i = Af 2, (25)

where A is a non-dimensional constant of integration.


The physical quantities of interest are the skin friction coefficient and the Stanton number, which aredefined as

Cf = �w

�U2∞, St = Qw

�U∞CpT ∗L, (26)

where �w is the skin friction and Qw is the total heat transfer from the plate which are given as

�w =[(� + �)

�u

�y+ �N

]y=0

, Qw = �Cp

∫ ∞

0[(T − T∞)u]x=� dy, (27)

with � being the length of the plate and Cp is the specific heat capacity at constant pressure. Using variables(8) and (17), we get

Cf Re1/2x = [1 + (1 − n)K]f ′′(0), St Re1/2 =

∫ ∞

0f ′� d = constant, (28)

where Rex = U∞x/� is the local Reynolds number. According to Eq. (28) the Stanton number is inde-pendent of the plate length �, thereby confirming the statement that the total heat transfer takes place atthe leading edge.

Following Steinrück (2001), we consider further the behavior of the similarity solutions ofEqs. (18)–(20) and (23) for large values of the Prandtl number. In this case of large values of Pr(?1),the buoyancy effects are confined to a small temperature sub-layer of the boundary layer. Thus, we takethe following new similarity variables:

= Pr1/3, f () = Pr−2/3f (), �() = �(), h() = h(),i() = Pr−2/3i(), � = Pr2/3�, A = Pr2/3A, (29)

so that we get the similarity equations

2(1 + K)f ′′′ + 2Kh′ + �� = 0, (30)

(1 + K/2)(ih′)′ − K(2h + f ′′) = 0, (31)

f i′ − 2f ′i = 0, (32)

2�′ + f � = 0, (33)

subject to the boundary and matching conditions

f (0) = f ′(0) = i(0) = 0, �(0) = 1, h(0) = −nf ′′(0),f ′′(∞) = f ′′

m(0) = C(K), h(∞) = hm(0), (34)

for Pr → ∞, where for this case primes denote differentiation with respect to . Further, C(K) is aconstant, which depends on the material parameter K, and f ′′

m(0) and hm(0) are solutions of the Blasiusprofiles for the micropolar fluid, which are determined as follows. Since Pr?1, the temperature differencesare negligible in the main part of the boundary layer, and the temperature is equal to that of the ambientfluid. Thus following Steinrück (2001), the velocity and microrotation profiles are given within theboundary layer by the modified Blasius profile that is by the equations

2(1 + K)f ′′′m + fmf ′′

m + 2Kh′m = 0, (35)


2(1 + K/2)(imh′m)′ + im(fmh′

m + f ′mhm) − 2K(2hm + f ′′

m) = 0, (36)

fmi′m − 2f ′mim = 0, (37)

subject to the boundary conditions

fm(0) = f ′m(0) = im(0) = 0, hm(0) = −nf ′′

m(0),f ′

m(∞) = 1, hm(∞) = 0. (38)

The values of f ′′(0) are given for Pr → ∞ by f ′′(0) = d2f /d2|=0.

3. Results and discussion

The corresponding sets of ordinary differential equations (18), (19), (23) and (25) subjected to theboundary conditions (22), as well as Eqs. (30)–(33) and (35)–(37), subjected to boundary conditions (34)and (38), respectively, have been solved numerically using the Keller-box method as described in the bookby Cebeci and Bradshaw (1988) for different values of the Prandtl number Pr, buoyancy parameters �and �Q = �

∫∞0 f ′� d, and the material parameter K while the constant of integration A is fixed, namely

A= 1. The parameter�Q can be interpreted as a buoyancy parameter, which is based on the enthalpy fluxthrough a cross section of the boundary layer (Steinrück, 2001). We assume that gyration is taken to beequal to the angular velocity at the plate, which is a representative of weak concentrations, i.e. n = 1

2 .

The variation of the dimensionless skin friction coefficient Cf Re1/2x and Stanton number St Re1/2 as a

function of buoyancy parameter � are shown in Figs. 2 and 3, respectively, for Pr = 1 and some valuesof material parameter K , while the variation of Cf Re1/2

x with �Q(=� St Re1/2) for some values of Pr inthe range from Pr = 0.01 to Pr → ∞ when K = 1 is illustrated in Fig. 4. As can be seen from Fig. 2,a unique solution of the similarity equations is obtained for positive values of �. Also for � > 0 (platetemperature larger than free-stream temperature), there is a favorable pressure gradient above the platedue to buoyancy effects which results in the flow being accelerated in a larger skin friction coefficientthan in the non-buoyant case (�=0). For negative values of � (plate temperature smaller than free-streamtemperature), two solutions (�c < � < 0), unique solution (� = �c) or no solution (� < �c) is obtained,where �c represents the critical values of � as shown in Table 1. Numerical values of these solutions forskin friction coefficient and Stanton number are tabulated in Table 2 and Table 3, respectively. At �=�c,both solution branches are connected, thus a unique solution is obtained. Note that at � = �c the valuesof Cf Re1/2

x are positives for all K and Pr. Following the lower solution branch with increasing �, onecan see that the skin friction is decreasing and changes sign at � = �a. For �a < � < 0, the skin frictioncoefficient is negative thus indicating a reverse flow region. The second critical point is �=Cf Re1/2

x =0,which does not lie on the curve. The details for the case of Newtonian fluid (K = 0) can be found in thepaper by de Hoog et al. (1984).

Numerical results for the Stanton number are shown in Fig. 3. The interval of � for which dual solutionsexist are the same as skin friction, that is �c < � < 0, where the values of �c for some values of K andPr are given in Table 1 along with those reported by Schneider (1979) and Magyari et al. (2002), for thecase of K = 0 (Newtonian fluid). As can be seen in Fig. 3, St Re1/2 approaches +∞ as � approaches zerofrom the left. The second value of St Re1/2 for a particular value of K could not be obtained for � = 0,which is consistent with the result of Cf Re1/2

x (see Fig. 2).


Fig. 2. Skin friction coefficient as a function of buoyancy parameter � for various values of material parameter K , when Pr = 1.

Fig. 3. Stanton number as a function of buoyancy parameter � for various values of material parameter K , when Pr = 1.


Fig. 4. Skin friction coefficient as a function of �Q = � St Re1/2 for various values of Prandtl number Pr, when K = 1.

Table 1Critical values of buoyancy parameter � (=�c) for various K and Pr

K Pr Present Schneider (1979) Magyari et al. (2002)

0 0.5 −0.0594 −0.05771 −0.0813 −0.0787 −0.0813592 −0.1139 −0.1099

1 0.01 −0.01180.1 −0.03201 −0.0852

10 −0.3008100 −1.2670

2 1 −0.08624 1 −0.0875

Figs. 5–7 present the velocity, angular velocity and temperature profiles, respectively, for various Kwhile � = −0.04 and Pr = 1. As can be observed from Figs. 2 and 3 as well as Tables 2 and 3, thereare dual solutions when � = −0.04. These figures support the validity of both solutions. The effects of Kon the velocity and temperature can be observed from Figs. 5 and 7, respectively, that increasing K is todecrease the velocity but to increase the temperature, while for angular velocity, the effect of K dependson the distance of the fluid from the plate, as shown in Fig. 6. It is worth mentioning that since thereare dual solutions for this value of �, there are two curves for a particular value of K . Both curves arecoinciding when � = �c as can be seen in Fig. 8 for � = −0.0852.


Fig. 5. Velocity profiles for various K when Pr = 1 and � = −0.04.

Fig. 6. Angular velocity profiles for various K when Pr = 1 and � = −0.04.


Fig. 7. Temperature profiles for various K when Pr = 1 and � = −0.04.

Fig. 8. Temperature profiles for various � when Pr = 1 and K = 1.


Table 2Values of Cf Re1/2

x for various � and Pr when K = 1

� Pr = 0.1 Pr = 1 Pr = 10

First solution Second solution First solution Second solution First solution Second solution

−0.3 0.1492 0.1220−0.2 0.2946 0.0056−0.1 0.3653 −0.0290−0.08 0.2001 0.0741 0.3774 −0.0322−0.06 0.2838 0.0108 0.3889 −0.0338−0.05 0.3128 −0.0070 0.3946 −0.0339−0.04 0.3383 −0.0202 0.4001 −0.0333−0.02 0.2907 −0.0017 0.3827 −0.0349 0.4109 −0.0290

0 0.4214 0.4214 0.42140.2 0.5167 0.6928 0.51300.4 1.4546 0.8874 0.58970.5 1.6283 0.9718 0.6246

Table 3Values of St Re1/2 for various � and Pr when K = 1

� Pr = 0.1 Pr = 1 Pr = 10

First solution Second solution First solution Second solution First solution Second solution

−0.3 0.2819 0.2766−0.2 0.3093 0.2672−0.1 0.3214 0.3354−0.08 1.3027 1.2232 0.3235 0.3777−0.06 1.3581 1.2196 0.3253 0.4511−0.05 1.3753 1.2490 0.3262 0.5105−0.04 1.3909 1.3102 0.3271 0.5997−0.02 5.1604 4.8827 1.4173 1.6893 0.3289 1.0328

0 5.1659 1.4396 0.33050.2 5.5134 1.5780 0.34430.4 6.6302 1.6626 0.35500.5 6.8006 1.6964 0.3597

From Figs. 7 and 8, it is observed that �′(0) = 0 for all values of K and �, respectively. These resultsindicate that there is no local heat transfer at the plate surface for all x > 0. However, the temperatureof the fluid is changed during the flow process. Thus, there must be a singular behavior of the walltemperature at x = 0, see Eq. (16). Therefore, all the heat necessary to change the fluid temperature mustbe transferred in the singular point x =0, which is the leading edge of the plate. In Fig. 4, the skin frictioncoefficient as a function of �Q = �

∫∞0 f ′� d is plotted for Prandtl numbers in the range from 0.01 to

∞. The parameter�Q can be interpreted as a buoyancy parameter, which is based on the enthalpy fluxthrough a cross section of the boundary layer. Since there is no heat transfer at the wall, the enthalpy fluxthrough a cross section is constant. In the limiting case Pr → ∞, the skin friction behaves qualitatively


similar to the case of other Prandtl numbers. According to the second equation of Eq. (28), the Stantonnumber is independent of the plate length, thereby confirming the statement that the total heat transfertakes place at the leading edge of the plate.

4. Conclusions

In this paper, we have theoretically studied the mixed convection flow and heat transfer over a horizontalplate in a micropolar fluid by assuming that the wall temperature is inversely proportional to the square rootof the distance from the leading edge. The governing boundary layer equations were solved numericallyusing the Keller-box method. The development of velocity profiles, angular velocity or microrotationprofiles and temperature profiles as well as skin friction coefficient and Stanton number have beenillustrated in tables and figures. A discussion for the effects of buoyancy parameter �, material parameterK and Prandtl number Pr on the skin friction coefficient Cf Re1/2

x and Stanton number St Re1/2 for the casen= 1

2 (weak concentration particles at the plate) has been done. It is found that the set of similarity equationshas dual solutions, unique solution or no solution, depending on the value of buoyancy parameter �, forall material parameter K and Prandtl number Pr. Moreover, it is found that although the wall temperatureis different from the free stream temperature, there is no local heat transfer at the wall except in thesingular point at the leading edge, which is similar to the case of Newtonian fluid.

Acknowledgements

The authors gratefully acknowledged the financial support received in the form of a research grant(IRPA project code: 09-02-02-10038-EAR) from the Ministry of Science, Technology and Innovation(MOSTI), Malaysia. They also wish to express their sincere thanks to the anonymous referees for theirvaluable comments and suggestions.

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The Schneider problem for a micropolar fluid